1

Describe, analyze and generalize patterns arising in a variety of contexts and express them in general terms. Bm1

2

Represent and record patterns in a variety of ways including tables, charts and graphs, and translate between various representations.Bm2

3

Use patterns and their generalizations to make and justify inferences and predictions. Bm3

4

Explore and describe visual and numeric patterns, including linear expressions, near-linear patterns, and symmetric and spatial patterns.Bm4

5

Use patterns and generalizations to solve problems and explore new content. Bm5

6

Identify and describe the nature of change; recognize change in more abstract and complex situations and explore different kinds of change, and patterns of variation. Bm6

7

Connect an initial state to a final state and generalize a rule that describes a pattern of change. Bm7

8

Begin to investigate applications in bivariate data and linear relationships, and explore questions of what will happen to one quantity if another variable is changed. Bm8

9

Represent variability or change by ordered pairs, tables, graphs and equations. Bm9

10

Differentiate between functions and relationships such as linear vs. not linear or continuous vs. non-continuous. Bm10

11

Continue to explore relationships arising from interesting contexts, and use variables and relationships to solve mathematical problems. Bm11

12

Select and use appropriate tools; measure objects using standard units in both the metric and common systems, and measure angles in degrees. Bm12

13

Identify the attribute to be measured and select the appropriate unit of measurement for length, mass (weight), time, temperature, perimeter, area, volume, angle. Bm13

14

Estimate measures with a specified degree of accuracy and decide if an estimate or a measurement is “close enough.” Bm14

15

Interpret measurements and recognize that two objects may have the same measurement on one attribute (e.g., area), but not necessarily on another (e.g., perimeter).  Bm15

16

Use proportional reasoning and indirect measurements to draw inferences. Bm16

17

Apply measurement to describe the real world and to solve problems. Bm17

18

Locate and describe objects in terms of their position, including compass directions, Cartesian coordinates, latitude and longitude, and midpoints. Bm18

19

Locate and describe objects in terms of their orientation and relative position, including coincident, collinear, parallel, perpendicular; differentiate between fixed (e.g., N-S-E-W) and relative (e.g., right-left) orientations; recognize and describe examples of bilateral and rotational symmetry. Bm19

20

Describe translations, reflections, rotations and dilations using the language of transformations, and employ transformations to verify congruence of figures. Bm20

21

Locate the position of points or objects described by two or more conditions; locate all the points (locus) that satisfy a given condition.  Bm21

22

Use concepts of position, direction and orientation to describe the physical world and to solve problems. Bm22

23

Distinguish among shapes and differentiate between examples and non-examples of shapes based on their properties; generalize about shapes of graphs and data distributions. Bm23

24

Organize data using tables, charts, graphs, spreadsheets and databases. Bm24

25

Construct familiar shapes using coordinates, appropriate tools (including technology), sketching and drawing two- and three- dimensional shapes. Bm25

26

Combine, dissect and transform shapes. Bm26

27

Generalize about the common properties of similar, congruent, parallel and perpendicular shapes and verify their generalizations informally. Bm27

28

Use shape, shape properties and shape relationships to describe the physical world and to solve problems. Bm28

29

Collect and explore data through observation, measurement, surveys, sampling techniques and simulations.  Bm29

30

Organize data using tables, charts, graphs, spreadsheets and databases. Bm30

31

Present data using a variety of appropriate representations, and explain why one representation is preferred over another or how a particular representation may bias the presentation. Bm31

32

Identify what data are needed to answer a particular question or solve a given problem, and design and implement strategies to obtain, organize and present those data. Bm32

33

Critically read data from tables, charts or graphs and explain the source of the data and what the data represent. Bm33

34

Describe the shape of a data distribution and identify the center, the spread, correlations, and any outliers. Bm34

35

Draw, explain and justify conclusions based on data. Bm35

36

Critically question the sources of data; the techniques used to collect, organize and present data; the inferences drawn from the data; and the possible sources of bias in the data or their presentation. Bm36

37

Formulate questions and problems, and gather and interpret data to answer those questions. Bm37

38

Develop an understanding of integers and rational numbers, and represent rational numbers in both fraction and decimal form. Bm38

39

Extend their understanding of numeration systems to include decimal numeration, scientific numeration and non-decimal numeration systems. Bm39

40

Develop an understanding of the properties of the integer and rational number systems (e.g., order, density) and of the properties of special numbers including 0, 1, p, and the additive and multiplicative inverses. Bm40

41

Apply their understanding of number systems to model and solve mathematical and applied problems. Bm41

42

Compare and order integers and rational numbers using relations of equality and inequality.  Bm42

43

Express numerical comparisons as ratios and rates.  Bm43

44

Distinguish between prime and composite numbers; identify factors, multiples, common factors and multiples, and relatively prime numbers; and apply divisibility tests to numbers. Bm44

45

Explain the meaning of powers and roots of numbers and use calculators to compute powers and square roots. Bm45

46

Apply their understanding of number relationships in solving problems. Bm46

47

Give geometric representations of fractions, prime and composite numbers, triangular and square numbers, and other number concepts; represent rational numbers and integers on the number line.  Bm47

48

Recognize equivalent representations of a number, especially fractions, decimals and percents, and translate freely among representations. Bm48

49

Distinguish between numbers that are used for counting, numbers that are used for ordering, numbers that are used for measuring, and numbers that are used for naming. Bm49

50

Develop and refine strategies for estimating quantities, including fractional quantities, and evaluate the reasonableness and appropriateness of their estimates. Bm50

51

Select appropriate representations for numbers, including integers and rational numbers, in order to simplify and solve problems. Bm51

52

Read and write algebraic expressions; develop original examples ex-pressed verbally and algebraically; simplify expressions and translate between verbal and algebraic expressions; and solve linear equations and inequalities. Bm52

53

Represent algebraic concepts with geometric models (e.g., algebra tiles), physical models (e.g., balance beam), tables and graphs; and write algebraic expressions to correspond to the multiple representations. Bm53

54

Solve linear equalities and inequalities using algebraic and geometric methods, and use the context of the problem to interpret and explain their solutions. Bm54

55

Analyze problems modeled by linear functions, determine strategies for solving the problems, and evaluate the adequacy of the solutions in the context of the problems. Bm55

56

Explore problems that reflect the contemporary uses of mathematics in significant contexts and use the power of technology and algebraic and analytic reasoning to experience the ways mathematics is used in society. Bm56

57

Use manipulatives and diagrams to model operations and their inverses with integers and rational numbers and relate the models to their symbolic expressions.  Bm57

58

Extend their understanding of numeration systems to include decimal numeration, scientific numeration and non-decimal numeration systems. Bm58

59

Describe the properties of operations with rationals and integers (e.g., closure; associative, commutative and distributive properties) and give examples of how they use those properties. Bm59

60

Efficiently and accurately apply operations with integers, rational numbers, and simple algebraic expressions in solving problems. Bm60

61

Use manipulatives, diagrams, and the fundamental theorem of counting to count permutations and combinations. Bm61

62

Use sets and set relationships to explore and solve simple algebraic and geometric problems. Bm62

63

Solve problems involving networks, for example planning delivery routes or counting paths between points. Bm63

64

Explore recurrence relations and iterations. Bm64

65

Continue to use manipulatives and drawings to model the concepts and procedures for the standard arithmetic algorithms, and develop and analyze their own and other students’ algorithms to accomplish a task or solve a mathematical problem. Bm65

66

Use discrete mathematics concepts as described above to model situations and solve problems; and look for whether or not there is a solution (existence problems), determine how many solutions there are (counting problems), and decide upon a best solution (optimization problems). Bm66

67

Describe events as likely or unlikely and give qualitative and quantitative descriptions of the degree of likelihood. Bm67

68

Describe probability as a measure of certainty ranging from 0 to 1, and conduct activities that allow them to express probabilities of simple events in mathematical terms. Bm68

69

Conduct experiments and give examples to illustrate the difference between dependent and independent events. Bm69

70

Explain the difference between probabilities determined from experiments or chance events (empirical) and probabilities derived mathematically (theoretical), and explain how the empirical probability changes for a large number of trials. Bm70

71

Conduct probability experiments and simulations to model and solve problems. Bm71